Is there a way to set the radius of an edge loop exactly by typing in a number somewhere?
If know the current diameter of the loop, you can scale the whole loop by x/d, where x is your new diameter, and d is your current one.

To determine the current diameter, you can use the Measure tool in the 3D viewport. Set the snapping mode to Vertex and Click+drag between two vertices on opposite sides of the loop. (It appears that even with snapping enabled, you still have to hold the Ctrl key to snap with the Measure tool.)



Once you know the current diameter of the loop, select the whole loop and scale by x/d.

For instance, if you want the final diameter to be 6.5, and your current diameter is 3.60979 (as in the example above), you can scale the loop by 6.5/3.60979, or 1.80066. Press S to scale, and type the value.

Note: Because of the way Blender parses math during operations, you may not be able to type 6.5/3.60979 during the operation. You either have to calculate the value beforehand and type the one number (1.80066), or you can scale twice, once by 6.5 and once by 1/3.60979.